Question: Simplify and expand the following expression: $ \dfrac{2}{t + 2}+\dfrac{t}{5t + 9} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(t + 2)(5t + 9)$ Multiply the first term by $\dfrac{5t + 9}{5t + 9}$ $ \begin{align*} \dfrac{2}{t + 2} \times \dfrac{5t + 9}{5t + 9} & = \dfrac{(2)(5t + 9)}{(t + 2)(5t + 9)} \\ & = \dfrac{10t + 18}{(t + 2)(5t + 9)}\end{align*} $ Multiply the second term by $\dfrac{t + 2}{t + 2}$ $ \begin{align*} \dfrac{t}{5t + 9} \times \dfrac{t + 2}{t + 2} & = \dfrac{(t)(t + 2)}{(5t + 9)(t + 2)} \\ & = \dfrac{t^2 + 2t}{(5t + 9)(t + 2)}\end{align*} $ Now we have: $ = \dfrac{10t + 18}{(t + 2)(5t + 9)} + \dfrac{t^2 + 2t}{(5t + 9)(t + 2)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{10t + 18 + t^2 + 2t}{(t + 2)(5t + 9)} $ $ = \dfrac{12t + 18 + t^2}{(t + 2)(5t + 9)}$ Expand the denominator: $ = \dfrac{12t + 18 + t^2}{5t^2 + 19t + 18}$